Quantum Matter

Dataset augmentation, the practice of applying a wide array of domain-specific transformations to synthetically expand a training set, is a standard tool in supervised learning. While effective in tasks such as visual recognition, the set of transformations must be carefully designed, implemented, and tested for every new domain, limiting its re-use and generality. In this talk, I will describe recent methods that  transform data not in input space, but in a feature space found by unsupervised learning. We start with data points mapped to a learned feature space and apply simple transformations such as adding noise, interpolating, or extrapolating between them. Working in the space of context vectors generated by sequence-to-sequence recurrent neural networks, this simple and domain-agnostic technique is demonstrated to be effective for both static and sequential data. 

Bio: Graham Taylor is an Associate Professor at the University of Guelph where he leads the Machine Learning Research Group. He is a member of the Vector Insitute for Artificial Intelligence and is an Azrieli Global Scholar with the Canadian Institute for Advanced Research. He received his PhD in Computer Science from the University of Toronto in 2009, where he was advised by Geoffrey Hinton and Sam Roweis. He spent two years as a postdoc at the Courant Institute of Mathematical Sciences, New York University working with Chris Bregler, Rob Fergus, and Yann LeCun. 

Dr. Taylor's research focuses on statistical machine learning, with an emphasis on deep learning and sequential data. Much of his work has focused on "seeing people" in images and video, for example, activity and gesture recognition, pose estimation, emotion recognition, and biometrics.

The hydrodynamic approximation is an extremely powerful tool to describe the behavior of many-body systems such as gases. At the Euler scale (that is, when variations of densities and currents occur only on large space-time scales), the approximation is based on the idea of local thermodynamic equilibrium: locally, within fluid cells, the system is in a Galilean or relativistic boost of a Gibbs equilibrium state. This is expected to arise in conventional gases thanks to ergodicity and Gibbs thermalization, which in the quantum case is embodied by the eigenstate thermalization hypothesis. However, integrable systems are well known not to thermalize in the standard fashion. The presence of infinitely-many conservation laws preclude Gibbs thermalization, and instead generalized Gibbs ensembles emerge. In this talk I will introduce the associated theory of generalized hydrodynamics (GHD), which applies the hydrodynamic ideas to systems with infinitely-many conservation laws. It describes the dynamics from inhomogeneous states and in inhomogeneous force fields, and is valid both for quantum systems such as experimentally realized one-dimensional interacting Bose gases and quantum Heisenberg chains, and classical ones such as soliton gases and classical field theory. I will give an overview of what GHD is, how its main equations are derived, its relation to quantum and classical integrable systems, and some geometry that lies at its core. I will then explain how it reproduces the effects seen in the famous quantum Newton cradle experiment, and how it leads to exact results in transport problems such as Drude weights and non-equilibrium currents.

This is based on various collaborations with Alvise Bastianello, Olalla Castro Alvaredo, Jean-Sébastien Caux, Jérôme Dubail, Robert Konik, Herbert Spohn, Gerard Watts and my student Takato Yoshimura, and strongly inspired by previous collaborations with Denis Bernard, M. Joe Bhaseen, Andrew Lucas and Koenraad Schalm.

Frustrated magnets provide a fertile ground for discovering exotic states of matter, such as those with topologically non-trivial properties. Motivated by several near-ideal material realizations, we focus on aspects of the two-dimensional kagome antiferromagnet. I present two of our works in this area both involving the spin-1/2 XXZ antiferromagnetic Heisenberg model. First, guided by a previous field theoretical study, we explore the XY limit ($J_z=0$) for the case of 2/3 magnetization (i.e. 1/6 filling of hard-core bosons) and perform exact numerical computations to search for a "chiral spin liquid phase". We provide evidence for this phase by analyzing the energetics, determining minimally entangled states and the associated modular matrices, and evaluating the many-body Chern number [1]. The second part of the talk follows from an unexpected outcome of the first work, which realized the existence of an exactly solvable point for the ratio of Ising to transverse coupling $J_z/J=-1/2$. This point in the phase diagram has "three coloring" states as its exact ground states, exists for all magnetizations (fillings) and is found to be the source or "mother" of the observed phases of the kagome antiferromagnet. Using this viewpoint, I revisit certain aspects of the highly contentious Heisenberg case (in zero field) and suggest that it is possibly part of a line of critical points.

[1] K. Kumar, H. J. Changlani, B. K. Clark, E. Fradkin, Phys. Rev. B 94, 134410 (2016)
[2] H. J. Changlani, D. Kochkov, K. Kumar, B. K. Clark, E. Fradkin, under review.

Many model quantum spin systems have been proposed to realize critical points or phases described by 2+1 dimensional conformal gauge theories. On a torus of size L and modular parameter τ, the energy levels of such gauge theories equal (1/L) times universal functions of τ. We compute the universal spectrum of QED3, a U(1) gauge theory with Nf two-component massless Dirac fermions, in the large-Nf limit. We also allow for a Chern-Simons term at level k, and show how the topological k-fold ground state degeneracy in the absence of fermions transforms into the universal spectrum in the presence of fermions; these computations are performed at fixed Nf/k in the large-Nf limit.

A fundamental assumption of quantum statistical mechanics is that closed isolated systems always thermalize under their own dynamics. Progress on the topic of many-body localization has challenged this vital assumption, describing a phase where thermalization, and with it, equilibrium thermodynamics, breaks down.

In this talk, I will describe how we can realize such a phase of matter with ultracold fermions in both driven and undriven optical lattices, with a focus on the relevance of realistic experimental platforms. Furthermore, I will describe new results on the observation of a regime exhibiting extremely slow scrambling, even for "infinite-temperature states" in one and two dimensions. Our results demonstrate how controlled quantum simulators can explore fundamental questions about quantum statistical mechanics in genuinely novel regimes, often not accessible to state-of-the-art classical computations.

We classify quantum states proximate to the semiclassical Neel state of the spin S=1/2 square lattice antiferromagnet with two-spin near-neighbor and four-spin ring exchange interactions. Motivated by a number of recent experiments on the cuprates and the iridates, we examine states with Z_2 topological order, an order which is not present in the semiclassical limit. Some of the states break one or more of reflection, time-reversal, and lattice rotation symmetries, and can account for the observations. We discuss implications for the pseudogap phase.

How can we quantify the entanglement between subsystems when we only have access to incomplete information about them and their environment?‎ Existing approaches (such as Rényi entropies) can only detect the short-range entanglement across a boundary between a subsystem and its surroundings, and then only if the whole system is pure. These methods cannot detect the long-range entanglement between two subsystems embedded in a larger system. There is a natural choice of entanglement measure for this situation, called the entanglement negativity, which can do this and cope with mixed states as well. However it is defined in terms of the full density matrix, which we generally won't have access to.

I will begin this talk with a brief overview ‎of some replica trick-based eigenspectrum reconstruction methods, and their various strengths and limitations. Then I will show how to modify these to find the moments of the partially transposed density matrix. Once those numbers have been obtained, it is possible to modify the earlier eigenspectrum reconstruction methods to obtain lower and upper bounds for the entanglement negativity.

Addendum: An audience member pointed out that the adjective "quasi-topological" already has a meaning and it's something different from the subject of this talk. So with hindsight it should have been called `quasi-conformal quantum computing'

Recent studies of highly frustrated antiferromagnets (AFMs) have demonstrated the qualitative impact of virtual, longer-range singlet excitations on the effective RVB tunneling parameters of the low energy sector of the problem [1,2]. Here, I will discuss the current state of affairs on the RVB description of the spin-1/2 kagome AFM, and present new results that settle a number of issues in this problem [3].

[1] I. Rousochatzakis, Y. Wan, O. Tchernyshyov, and F. Mila, PRB 90,

100406(R) (2014)

[2] A. Ralko and I. Rousochatzakis, PRL 115, 167202 (2015) [3] in preparation.

We consider the problem of certifying entanglement and nonlocality in one-dimensional translation-invariant (TI) infinite systems when just averaged near-neighbor correlators are available. Exploiting the triviality of the marginal problem for 1D TI distributions, we arrive at a practical characterization of the near-neighbor density matrices of multi-separable TI quantum states. This allows us, e.g., to identify a family of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that, when viewed as a vector in R^n, the set of boxes admitting an infinite TI classical extension forms a polytope, i.e., a convex set defined by a finite number of linear inequalities. Using DMRG, we prove that some of these inequalities can be violated by distant parties conducting identical measurements on an infinite TI quantum state. Both our entanglement witnesses and our Bell inequalities can be used to certify entanglement and nonlocality in large spin chains (namely, finite, and not TI chains) via neutron scattering.

Our attempts at generalizing our results to TI systems in 2D and 3D lead us to the virtually unexplored problem of characterizing the marginal distributions of infinite TI systems in higher dimensions. In this regard, we show that, for random variables which can only take a small number of possible values (namely, bits and trits), the set of nearest (and next-to-nearest) neighbor distributions admitting a 2D TI infinite extension forms a polytope. This allows us to compute exactly the ground state energy per site of any classical nearest-neighbor Ising-type TI Hamiltonian in the infinite square or triangular lattice. Remarkably, some of these results also hold in 3D.

In contrast, when the cardinality of the set of possible values grows (but remaining finite), we show that the marginal nearest-neighbor distributions of 2D TI systems are not described by a polytope or even a semi-algebraic set. Moreover, the problem of computing the exact ground state energy per site of arbitrary 2D TI Hamiltonians is undecidable.

Quantum triangles can work as interferometers. Depending on their geometric size and interactions between paths, “beats” and/or “steps”

patterns are observed. We show that when inter-level distances between level positions in quantum triangles periodically change with time, formation of beats and/or steps no longer depends only on the geometric size of the triangles but also on the characteristic frequency of the transverse signal. For large-size triangles, we observe the coexistence of beats and steps for moderated frequencies of the signal and for large frequencies a maximum of four steps instead of two as in the case with constant interactions are observed.

Small-size triangles also revealed counter-intuitive interesting dynamics for large frequencies of the field: unexpected two-step patterns are observed. When the frequency is large and tuned such that it matches the uniaxial anisotropy, three-step patterns are observed.

We have equally observed that when the transverse signal possesses a static part, steps maximize to six. These effects are semi-classically explained in terms of Fresnel integrals and quantum mechanically in terms of quantized fields with a photon-induced tunneling process. Our expressions for populations are in excellent agreement with the gross temporal profiles of exact numerical solutions. We compare the semi-classical and quantum dynamics in the triangle and establish the conditions for their equivalence.

A central theme of modern condensed matter physics is the study of topological quantum matter enabled by quantum mechanics, which provides a further "topological" twist to the classical theory of ordered phases. These quantum-entangled phases of matter such as fractional quantum Hall phases, spin liquids, and some non-Fermi liquids, are typically strongly-correlated and thus cannot be studied within conventional perturbative approaches. Because of the spectacular emergent phenomena as well as their potential for realistic applications, there has been much recent interest in exploring the physics of these exotic phases. In this talk, I show that the powerful methods of quantum field theory, namely quantum anomaly and duality, can expose the global landscape in parameter space of these gapped and gapless topological quantum phases and lead to several novel insights on these phases. As a demonstration of this principle, we study clean fractional quantum Hall transitions, composite Fermi liquids, and the surface of fractional topological insulators. Despite long and storied histories, these three systems are at the frontier of our knowledge of two and three dimensional topological phases. I show that the non-perturbative approach for these systems, i.e., the duality, sheds some new light on these systems and allows us to resolve some longstanding puzzles, which have not been clear previously. Furthermore, it uncovers novel physics of these intrinsically strongly-correlated phases of matter.

The quest for quantum spin liquids is an important enterprise in strongly correlated physics, yet candidate materials are still few and far between. Meanwhile, the classical front has had far more success, epitomized by the exceptional agreement between theory and experiment for a class of materials called spin ices. It is therefore natural to introduce quantum fluctuations into this well-established classical spin liquid model, in the hopes of obtaining a fully quantum spin liquid state.

The spin-flip excitations in spin ice fractionalize into pairs of effective magnetic monopoles of opposite charge. Quantum fluctuations have a parametrically larger effect on monopole motion than on the spin ice ground states so the leading manifestations of quantum behavior appear when monopoles are present. We study magnetic monopoles in quantum spin ice, whose dynamics is induced by a transverse field term. For this model, we find a family of extensively degenerate excited states, that make up an approximately flat band at the classical energy of the nearest neighbor monopole pair. These so-called mesonic states are exact up to the splitting of the spin ice ground state manifold. In my talk I will discuss their construction and properties that may be relevant in neutron scattering experiments on quantum spin ice candidates.